Question

The true stress (in MPa) versus true strain relationship for a metal is given by

\(\sigma = 1020{\varepsilon ^{0.4}}\)

The cross-sectional area at the start of a test (when the stress and strain values are equal to zero) is 100 mm². The cross-sectional area at the time of necking (in mm²) is ________ (correct to two decimal places)

Answer 1

Concept:

The relationship between true stress and strain is governed by the equation 

σ = σ0 ϵη

At UTS, ϵ = n

ϵ = True strain \(= \ln \left( {\frac{{{L_f}}}{{{L_i}}}} \right)\)

Where (Lf) is the final length; (Li) is the initial length.

Calculation:

Here σ = 1020 ε^0.4

So ϵ = 0.4 at (UTS)

\(\Rightarrow 0.4 = \ln \left( {\frac{{{L_f}}}{{{L_i}}}} \right) \Rightarrow \frac{{{L_f}}}{{{L_i}}} = {e^{0.4}}\)

\(\Rightarrow \frac{{{L_f}}}{{{L_i}}} = 1.491\)

As volume remains constant throughout

(A_iL_i = A_f L_f) Given A_i = 100 mm²

⇒ 100 L_i = A_f L_f

\(\Rightarrow {A_f} = 100 \times \frac{{{L_i}}}{{{L_f}}} \Rightarrow {A_f} = \frac{{100}}{{1.49}}\)

A_f = 67.03 mm²